Preheat the oven to 350°F (175°C). Grease a 9-inch round cake pan and line the bottom with parchment paper.
Add the chopped apples to a bowl. Toss them with brown sugar, cinnamon, and melted butter until evenly coated. Set aside.
In a medium bowl, whisk together the flour, baking powder, cinnamon, nutmeg, and salt.
In a large mixing bowl, beat the softened butter, granulated sugar, and brown sugar until light and creamy.
Add the eggs one at a time, mixing well after each addition. Mix in the vanilla extract.
Add about half of the dry ingredients and mix gently. Add the milk and sour cream, followed by the remaining dry ingredients. Mix just until combined.
Spread the batter evenly into the prepared cake pan.
Spoon the cinnamon apple mixture evenly over the top. Gently press some of the apple pieces slightly into the batter while leaving plenty visible on the surface.
Bake for 45 to 55 minutes, or until the cake is golden brown and a toothpick inserted into the center comes out clean or with a few moist crumbs.
Allow the cake to cool in the pan for 15 minutes before transferring it to a serving plate.
Whisk together the powdered sugar, milk, and vanilla until smooth. Once the cake has cooled slightly, drizzle the glaze generously over the apples and allow some to run down the sides.